Conjugate heat transfer of Al2O3–water nanofluid in a square cavity heated by a triangular thick wall using Buongiorno’s two-phase model

  • A. I. Alsabery
  • T. Armaghani
  • A. J. Chamkha
  • I. Hashim


The present study investigates the conjugate heat transfer in a square cavity heated by a triangular solid and saturated with \(\text{Al}_2\text{O}_3\)–water nanofluid. Two-phase Buongiorno’s model is used for modeling the nanofluid heat transfer. The finite element method is used for numerical solution of the dimensionless governing equations subject to the boundary conditions. Comparisons of the proposed method with previously published experimental and numerical works show a good agreement. The effects of some parameters such as the Rayleigh number, thermal conductivity ratio, dimensionless triangular wall thickness and nanofluid volume fraction on heat transfer and nanoparticle distributions are completely studied and discussed. The results show clockwise rotations for streamlines and nanoparticle migration. Also the Nusselt number increases with the nanofluid volume fraction. A continuous reduction is seen for the mean Nusselt number by increasing the dimensionless triangular wall thickness for all the considered values of the Rayleigh number.


Nanofluid Heat transfer Brownian motion Thermophoresis effect Nanoparticle distribution Triangular solid 

List of symbols


Specific heat capacity (J kg\(^{1}\) K\(^{1}\))


Width and height of triangular solid wall (m)


Diameter of the base fluid molecule (nm)


Diameter of the nanoparticle (nm)


Dimensionless triangular wall thickness, \(D=d/L\)


Brownian diffusion coefficient (kg m\(^{-1}\) s\(^{-1}\))


Reference Brownian diffusion coefficient (kg m\(^{-1}\) s\(^{-1}\))


Thermophoretic diffusivity coefficient (kg m\(^{-1}\) s\(^{-1}\))


Reference thermophoretic diffusion coefficient (kg m\(^{-1}\) s\(^{-1}\))


Gravitational acceleration (ms\(^{-2}\))


Thermal conductivity (Wm\(^{-1}\) K\(^{-1}\))


Triangular wall to nanofluid thermal conductivity ratio, \(K_{\mathrm{r}}=k_{\mathrm{w}}/k_{\mathrm{nf}}\)


Width and height of enclosure (m)


Lewis number


Ratio of Brownian to thermophoretic diffusivity


Average Nusselt number


Prandtl number


Rayleigh number


Brownian motion Reynolds number


Temperature (K)


Reference temperature (310 K)


Freezing point of the base fluid (273.15 K)

\(\mathbf v\), \(\mathbf V\)

Velocity and dimensionless velocity vector (ms\(^{-1}\))


Brownian velocity of the nanoparticle (ms\(^{-1}\))

x, y and X, Y

Space coordinates and dimensionless space coordinates

Greek symbols


Thermal diffusivity (m\(^2\) s\(^{-1}\))


Thermal expansion coefficient (K\(^{-1}\))


Dimensionless temperature


Dynamic viscosity (kg m\(^-1\) s\(^-1\))


Kinematic viscosity (m\(^2\) s\(^{-1}\))


Density (kg m\(^{-3}\))


Solid volume fraction

\(\varphi ^*\)

Normalized solid volume fraction


Average solid volume fraction





Base fluid






Solid nanoparticles


Solid wall



The work was supported by the Universiti Kebangsaan Malaysia (UKM) research Grant DIP-2017-010. We thank the respected reviewers for their constructive comments which clearly enhanced the quality of the manuscript.

Supplementary material

10973_2018_7473_MOESM1_ESM.pdf (102 kb)
Supplementary material 1 (pdf 102 KB)


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Refrigeration & Air-conditioning Technical Engineering Department, College of Technical EngineeringThe Islamic UniversityNajafIraq
  2. 2.School of Mathematical Sciences, Faculty of Science & TechnologyUniversiti Kebangsaan MalaysiaBangiMalaysia
  3. 3.Department of Engineering, Mahdishahr BranchIslamic Azad UniversityMahdishahrIran
  4. 4.Department of Mechanical Engineering, Prince Sultan Endowment for Energy and the EnvironmentPrince Mohammad Bin Fahd UniversityAl-KhobarSaudi Arabia
  5. 5.RAK Research and Innovation CenterAmerican University of Ras Al KhaimahRas Al KhaimahUnited Arab Emirates

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